Function generator



y 30, 1.961 R. M. LICHTENSTEIN 2,986,704

FUNCTION GENERATOR Filed June 29, 1956 3 Sheets-Sheet l Inventor:fife/and ML/chtensteirv,

by M 2% His Attorrveg.

May 30, 1961 R. M. LICHTENSTEIN FUNCTION GENERATOR 3 Sheets-Sheet 3Filed June 29, 1956 PUL SE S OURC E PULSE SOURCE 75 Inventor" fPo/c-zndM. Lichtenstein,

log/26m 9 M H/s Attorney- United States Patent FUNCTION GENERATOR RolandM. Lichtenstein, Schenectady, N.Y., assignor to General ElectricCompany, a corporation of New York Filed June 29, 1956, Ser. No. 594,839

22 Claims. (Cl. 323-142) This invention relates to a pulse ratemeasuring apparatus which produces a signal which is a desired functionof input pulse rate. In addition, this invention relates to a class offunction generators utilizing this principle that converts an inputcurrent into an output current which is some prescribed function of theinput current.

In many engineering, scientific, and industrial areas, it becomesnecessary to measure the characteristic parameters of a time series ofevents. In practice, many different types of time series may occur.However, the simplest and most important of these are the periodicseries and the stationary random series. In the periodic series, thetime interval between any two consecutive events has the same fixedduration T. The quantity is called the rate. A random series, on theother hand, is characterized by the following property: The probabilitythat an event occurs in the short time interval t-t+dt is rdt, Where ris a characteristic parameter of the series similarly called the rate.If the rate for a random series is a constant, the random series iscalled stationary. In such a stationary random series, there occur, onthe average, rt events during a time interval of duration t, or r eventsper unit time. However, the rate r may depend on time, in which case therandom series is, technically, no longer stationary. However, the randomseries may be regarded as stationary for all practical purposes, as longas the rate does not vary too rapidly with time, i.e., as long as r a!is small as compared to r.

In detecting and measuring nuclear radiation, for example, random timeseries appear and are quite significant. That is, the radiation isdetected by means of counters, such as pulse chambers, proportionalcounters, or scintillation counters, which produce individual outputpulses in response to the radiation intensity representing the events inthe time series. When the radiation intensity does not vary with time,the random series is stationary. However, as pointed out before, even ifthe radiation intensity does vary with time, the random series may beregarded as stationary for all practical purposes if the rate does notvary too rapidly. In practice, this condition is often fulfilled and, asa consequence, the stationary random series is of practical importancein nuclear instrumentation.

Thus, industrial control systems for nuclear reactors require a devicethat measures the rate r of a time series. Such a device is normallydenominated as a counting rate meter. In a nuclear reactor duringthestart-up period, this rate r may vary over ma y many decades.Consequently, such control systems then require a counting rate devicewhose output current is proportional to thelogarithm of the rate, sothat the device is able to follow the radiation intensity without scaleswitching. Furthermore, the time derivative of the logarithm of theradiation intensity, the reciprocal of this quantity being called theperiod of the reactor, is of fundamental importance in reactor control.Thus, a combination of a counting rate meter with a logarithmic outputand a differentiating means constitutes a period meter. In View of thesecharacteristics-ability to handle large ranges and convertibility into aperiod meter-a counting rate meter with a logarithmic output is a highlyimportant and desirable device in the field of industrial controlsystems. v v p All hitherto available devices having a logarithmicresponse have utilized a thermionic diode as one of the elements. Suchdevices, however, are very hard to keep stable since they are quitesusceptible to errors due to diode drift. That is, voltage shifts of theoperating char-'- acteristics of the diode occur due to changes in thetube parameters, such as changes in the emission characteristics of thecathode. Consequently, devices of thistype are extremely unsatisfactorydue to the inherently unstable characteristics of the thermionic diodes.

In the field of computers, especially analog computers, it is extremelydesirable to provide function generator; which convert input currentinto an output current which is some prescribed function of the inputcurrent. Count ing rate devices whose output current is proportional toa desired function of the repetition rate'of pulsesj may be cascaded toprovide such function generators. Funetion generators which utilize theprinciples of counting rate devices are especially desirable in thecomputer'field if the counting rate devices do not utilize unstableelements such as thermionic diodes, and if they may be constructed ofsimple components that are relatively in expensive. 1

It is an object of this invention to provide a highly accurate countingrate apparatus of simple design constructed with simple components andrelatively inexpen' sive to manufacture.

A further object of this invention is to provide a count ing rate devicewith a logarithmic output which does not contain any unstable elements.7

Yet another object of this invention is to provides network having atransfer admittance which is a specified function of a complex frequencywhich represents aura put pulse rate. l 5 Still another object of thisinvention is to provide a general class of function generators whichproduce an output current which is a desired function of current.

Yet another object of this invention is to provide ail apparatus whichprovides a pulse output, the repetition rate of which is a desiredfunction of an input current.

In practicing the invention, a net-work is provid which is constructedof simple components such asjgresistors, capacitances, and induotances.'The network is so constructed, according to network synthesis,principles, that a current is caused to flow when randominput pulses areapplied thereto which is a desired function of the rate of occurrence ofthe pulses. That is, the transferee; mittance of the network is made adesired function of it complex frequency, in this case representingtwice t e pulse rate, so that a current flows which depends on the pulserate in the same manner as the admittance of the network depends on thecomplex frequency, rinspecifig, function depends on the design of thenetwork. By as plying successively positive and negative voltagestoitlii network in response to the occurrence of successive events in thetime series", an output current is produced which a desiredfunction ofthe'inputpulse rate.

' It is also possible by utilizing such a counting rate device inconjunction with other elements" to reverse the operation and produce apulse output whose repetition frequency is a desired function of aninput current. That is, the output from a pulse source having a variableand adjustable repetition rate is applied to'a counting rate apparatusincluding a network of the type described above, to produce an outputcurrent which is a desired function of the pulse repetition rate. Theoutput of the counting rate device is compared to the input current andthe repetition rate of the pulses produced by the .pulser is varieduntil the output current from the counting rate device equals .the inputcurrent. When this i equality is achieved, the repetition rate of theoutput pulses is a specified function of the input current. Adevice ofthis type may be denominated asan anti-counting rate apparatus since itproduces a controllable pulse repetition rate in response to an inputcurrent.

By cascading an anti-counting rate device and a counting rate apparatus,it is possible to produce a function generator which converts an inputcurrent into an output current which is a given desired function of theinput current. That is, the repetition rate of the pulses from ananti-counting rate apparatus may be made a desired function of an inputcurrent by controlling the nature of the network within this apparatus.These pulses are then applied to the input of a counting rate apparatuswhich supplies an output current which is a desired function of thisinput repetition rate by controlling the type of network within thisapparatus. Thus, the output current can be made a desired function ofthe input current. Since the function generator contains two networks,one in the anti-counting rate apparatus, and the other in the countingrate apparatus, it can be seen that a very large number of functiongenerators may be synthesized by varying the individual characteristicsof these networks. V The novel features which are believed to becharacteristic of this invention are set forth with particularity in theappended claims. The invention itself, however, both as to itsorganization and method of operation, together with further objects andadvantages thereof, may best be understood by reference to the followingdescription taken in connection with the accompanying drawings in which:

Fig. '1 is a schematic diagram of a network embodying the principles ofthis invention and constructed in accordance therewith;

'Fig. 2 is a schematic circuit diagram of a counting rate apparatusconstructed in accordance with the invention and which embodies anetwork such as is shown in Fig. 1;

Figs. 3a and 3b are block diagrams of the showing of Fig. 2 and isutilized to show the underlying theoretica considerations of the circuitof Fig. 2;

Figs. 4a, 4b and 4c are a series of graphs of the voltage and currentwave forms appearing in different portions of the circuit shown in Fig.3;

Fig. 5 is a showing, partially in block diagram form, of anti-countingrate apparatus constructed in accordance with this invention;

Fig. 6 is a showing, also partially in block diagram form, of a functiongenerator constructed in accordance with this invention and utilizingconstructions such as shown in Figs. 1 and 2 as portions thereof.

Referring now to Fig. 1, a network is provided which is characterized bythe fact that it will transform input pulses into an output currentwhich is a function of the rate of occurrence of the input pulses. Therelative magnitudes of the network forming components are such that thetransfer admittance of the network is a function of the complexfrequency which, in the instant case, is the pulse rate of the inputpulses.

; The transfer admittance of a network may be defined m the'followingmanner: if a two-terminal-pair network has an input voltage applied toone pair of terminals,

4 which will be denoted as the input terminals, and the output terminalsare shorted together, the current flowing in the output terminals willbe determined by the transfer admittance of the network. That is, assumethat a voltage v(t) which is a function of time is applied to the inputterminal and is given by: v(t) =V e where V and s are constant complexnumbers. The parameter s is called the complex frequency. Insteady-state A.C. analysis s is normally restricted to purely imaginaryvalues is), where w is the radian frequency. However, it is more usefulto permit s to be a general complex number rather than to restrict it topurely imaginary values. When the voltage v(t)=V e is applied to theinput, then in the short-circuited output there flows a current i(t)which as a function of time t will be given by i(t)=l e The ratio ofi(t) to v(t) is an admittance and is given by the equation I Y(s) T707)and is the transfer admittance of the network. It can be seen that thistransfer admittance is a function of the complex frequency s.

Fig. 1 shows a network 1 having a pair of input terminals 2 and a pairof output terminals 3. The network consists of n parallel branches,which for the sake of simplicity and for illustrative purposes, havebeen shown as branches 4, 5, 6, 7, and 8. Each branch consists of aseries connected resistance-capacitance combination such as R4C4, R5C5,etc.

The network is characterized by a transfer admittance which is a desiredfunction of the complex frequency where the complex frequency representsthe rate of occurrence of the input pulses applied between the terminals2, so that a current flows in the output terminals 3 and through a loadcircuit which is related to the rate of occurrence of the pulses in thesame manner as the transfer admittance. Thus, the network 1 may bedescribed by the fact that the time average T of the output currentdepends on the rate of occurrence of the input pulses. That is,

The precise functional relationship of the time average of the outputcurrent T and the rate of occurrence r of the pulses may be controlledby controlling the relative magnitude of the resistive and capacitivecomponents of the network. Thus if an output current is desired which,for example, is a logarithmic function of the input pulse rate, anetwork is synthesized in which the resistive and capacitive elements ofthe various branches are related according to the equation:

where R, T, are design parameters, and a is a dimensionless numberassumed to be greater than unity. Dimensions of R and T are those of aresistance and time respec tively. The exact manner in which theseformulae for the resistance and capacitance elements are derived will beshown in detail later when a rigorous mathematical analysis will bedisclosed. However, it must be pointed out that the synthesizing ofnetworks having desired admittance characteristics are techniques wellknown to those skilled in the art, and reference is made to Symposium onModern Network Synthesis, Polytechnic Institute of Brooklyn, New York(1952), which provides an excellent review of the techniques andprinciples of network synthesis.

In a similar fashion, it is possible to synthesize a network which hasan admittance that is a fractional power of the complex frequency swhere the complex frequency represents twice the rate of occurrence 2rof S111 1m log a Tsin 1ra10ga MP1 where A, a, T and a are designparameters and a is a dimensionless number larger than unity and a .islarger than and smaller than unity. Then Y (s) and, in this particularcase Y (2r), is approximately By properly choosing values of at in therange between 0 and unity, a network may be synthesized which producesan output current which is any desired fractional power of the inputpulse rate.

It is possible, of course, by utilizing network synthesis techniques tosynthesize many other types of networks having other functionalrelationships to the complex frequency and, in turn, to a pulserepetition rate than the specific ones disclosed above. Consequently,these examples are not to be considered as limiting, but merelyillustrate specific examples of the broader inventive concept.

Referring now to Fig. 2, there is disclosed a counting rate apparatuswhich includes, among other elements, a network of the type disclosed inFig. 1. The counting rate apparatus of Fig. 2 comprises a network whichis characterized by a transferadmittance which is a desired function ofthe complex frequency where this complex frequency represents apulserate, an electronic switch means which is adapted to be switchedsuccessively to positive and negative states by successive input pulsesrepresenting the events in a random time series, and unidirectionalconducting means coupled to the output of the network whereby a currentflows which is a desired function of the rate of occurrence of the inputpulses.

The switch means consists of a bi-stable multivibrator 20 which, as iswell known, possesses two conditions of stable equilibrium. Thebi-stable device 20 consists of two cross-coupled electron dischargedevices 21 and 22. The electron discharge device 21 is a vacuum triodewhich has its anode connected to a source of energizing voltage withrespect to ground B-+ through an anode resistance 23. The cathode isconnected to a source of reference potential such as ground through acathode resistance 25 which is by-passed for AC. by means of acapacitance 26. In a similar manner, the electron discharge device 22has its anode connected to a source of voltage 8+ through an anoderesistance 24 while its cathode is connected to a source of referencepotential such as ground by means of the same cathode resistance 25.Thus, the cathodes of the two electron discharge devices are connectedthrough the common cathode resistance 25 and the bypass capacitance 26to provide a suitable cathode bias for the grids of the electrondischarge devices.

The anode of the discharge device 21 is coupled to the control grid ofdischarge device 22 through a parallel resistance capacitance circuit 29and to ground through the grid leak resistance 30. Similarly, the anodeof discharge device 22 is coupled to the control grid of the electrondischarge device 21 through a parallel resistance capacitance circuit 27and then to ground through the grid leak resistance 28. Circuits of thistype are char- .acterized :by the :fact that they possess two conditionsof stable equilibrium. One of these conditions is when the dischargedevice 21 is conducting and the discharge device 22 is non-conducting;and the other when discharge device 22 is conducting and dischargedevice 21 is cut off. The circuit remains in one or the other of thesetwo conditions with no change until some action occurs which causes thenon-conducting tube to conduct. The tubes then reverse their functionsand remain in the new condition until another action occurs. .It isbecause of this characteristic of remaining in one state of equilibriumuntil an event occurs which reverses the situation and brings aboutanother state of equilibrium, that this circuit is known as a bi-stablecircuit.

In order to change the equilibrium states of the device, it isnecessary, as has been pointed out, that the conducting tube be causedto become non-conducting while the cut-off tube becomes conductive. Tothis end, it is necessary to inject negative triggering pulses into thecircuit in order to achieve the reversal of states. A double diode 31mounted in a single envelope provides the means for injecting triggeringpulses into the circuit. The diode 31 contains a common cathode member32 connected to an input terminal 35 to which are applied a series ofnegative pulses representing the random events in the time series whoserate of counting is to be measured. A pair of diode anode members 33 and34 are connected respectively to the anodes of the discharge devices 21and 22.

Consequently, negative pulses appearing at the input terminals 35 areapplied through the diodes 31 to the anode of the non-conducting tube,since the anode of the non-conducting tube and that of the diodeconnected to it is at a high positive potential relative to a cathode 32of the diode 31, thus permitting conduction of the diode. The negativepulse applied to the anode of the non-conducting tube is applied to thegrid of the conducting tube through the capacitance element of one ofthe parallel resistance-capacitance circuits 27 and 29. The conductingtube is thus cut off, by virtue of the negative pulse applied to itscontrol grid, causing its anode voltage to rise thereby raising the gridvoltage of the formerly non-conducting tube, causing to to becomeconductive. This condition then continues until the next negative-inputpulse comes in at which time the equilibrium condition will again bereversed in'a similar fashion. Thus, the electron discharge devices 21and 22 will be switched successively to conducting and non-conductingconditions by successive input pulses. As a result the anode voltage ofthe electron discharge device 22 will successively be switched to apositive condition when the discharge is not conducting and to anegative state when it is conducting.

Connected to the anode of the discharge device 22 is a network 10 of thetype shown in Fig. 1 which consists of a number of parallel branches,11, 12, -16- n, each of which consists of a series connected resistancecapacitance combination. Coupled to the output of the network 10 are apair of oppositely poled rectifiers 36 and 37 and a meter 38 to providea measure of the current iiowing through the network 10 and therectifiers. The network 10, as was pointed out previously, may beconstructed to have a transfer admittance which is a 1 I 7 function ofthe complex frequency which in this instance i represents a pulse rate.Thus, there flows in the output .of the network a current which is adesired .function of the input pulse rate.

In operation, successive input pulses between terminal 35 and groundcauses the plate voltage of 'the electron discharge device 22 to becomesuccessively positive and. negative, since it is successively caused tobe conducting and non-conducting, and apply to the network 10consecutive step voltages of positive and negative signs which have thesame rate of occurrence as do the input pulses to terminal 35 whichrepresent events in a time serie'i,-

T7 ,Thus, if network 10 is constructed according to the formula:

2 log a a T log a a" that the output current may be utilized for otherpurposes, such as controls, than actuating a meter.

The foregoing description and explanation of the measuring apparatusillustrated in Figs. 1 and 2 will be more easily understood and the fullscope of the inventive concepts established if the following theoreticalbasis is established.

In order to do so, a generalized form of the apparatus of Fig. 2 isshown as Figs. 3a and 3b and will be utilized in establishing thetheoretical basis. The circuit of Fig. 3 consists of:

(1) Two voltage generators 42 and 43 functioning equipment and operatingvoltages, which, as functions of the time t, are written as v(t) and-v(t) respectively;

(2) A stable linear network 40, active or passive, including twoterminal pairs of suitable transfer admittance e (3) Two synchronizedsingle-pole, double-throw switches 41 and 44, both thrown either to theupper or lower position simultaneously. These switches are actuated insuch a fashion that they change position whenever an event in timeseries occurs.

The network shall have a transfer admittance Y(s), where s is a complexfrequency. The implication of this statement may be seen most clearlywith reference to Fig. 3b. When a voltage W(t) that vanishes for t andis quite arbitrarily for t 0, is applied to the left hand or inputterminal of the previously quiescent network 40, the

current i(t) in the upper lead of the right hand or output terminal isgiven by the equation Here L- denotes the operation Take the inverseLaplace transform of, and L denotes the operation Take the Lalacetransform of. In detail, Equations 1 and 2 are i(t) f Ye W(s) e"ds 3Where here i= and the integral in Equation 3 is taken along a suitablecontour in the complex s plane, according to the teachings of thecalculus of Laplace transforms, a

common tool in the analysis of transients. Combining Equations 3 and 4gives the resultant equation "8 4b which show respectively theconfiguration of w(t) and i(t). As a consequence of this restriction,the Equation 5 takes the form f Ye :w(r)e'"drds=0 a if the function W(r)fulfills the condition The operation of the device shown in Fig. 3a isas follows: Assume that the voltage generators are turned on at the timet=0 (there is no loss of generality in this assumption), simultaneouslythe synchronized switches are thrown alternately to the plus positioninto the minus position whenever an event of the time series occurs. Theposition of the switches may be described by a switching function (t),which assumes the value plus one when the switches are in the plusposition, and the value minus one when the switches are in the minusposition. An example of such switching function is illustrated in Fig.4c. The steps in the switching function, of course. coincide with theevents of the time series.

The voltage that is presented to the input of the network 40 of Fig. 3ais then no longer v(t) but v(t)f(t). According to Equation 5 the currenti,,(t) in lead a" is given by the equation Furthermore, the current i(t)in lead b" is defined by These two equations may be combined into oneequation by rewriting them as combining (8) and (10) results in the formConsidering a large number or ensemble of identical devices of the typeshown in Fig. 3a, and assuming each member of the ensemble is excited byidentical voltage generators furnishing the voltages v(t) and -v(t), afinal generalization of the circuit may be achieved. The switches ofeach member of the ensemble are thrown according to a time series whichdiffers from member to member. However, all of the time series shall bestationary random series the same rate r for all members of theensemble. Consequently, the ensemble average denoted by angularbrackets-of the current i(t) is formed. Accordingly, Equation 11 assumesthe form since at any instant 1- there are, on the average, as manymembers of the ensemble with the switches in the plus position (f(-r)=+1) as there are members with the switches in the minus position (f(-r)=-l).

Furthermore, f(t)f('r) =+1 if an even number of switching occurredbetween the instance I and 'r and j(t)f(-r)=l if an odd number ofswitching occurred between the instance 1 and '1'. These two be combinedinto one by writing equations may Inserting Equation 13 and Equation 16into Equation 12 changes its form into e r] 1 u t(t) 2 Y(s)e form-)4; ed-rds (17 The integral over '1' that occurs in Equation 17 may betransformed as follows:

f v()2 sin M21 (1' t) e "1dr -r=0 Inserting Equation 18 into Equation 17provides form i(l) =A-B (19) Where It will now be shown that theexpression B vanishes. Making the substitution for one of theintegration variables in Equation 21 gives an expression for B whichstates 10 Changing the name of the integration variable a inton thefollowing form for the equation is achieved Furthermore, because of thefactor sin h(2r1-) which vanishes for 1'=O,

Comparing Equation 22 with Equation 6, and Equation 24 with Equation 7it can be seen that B=0 ('25-) Combining Equations 19, 20, and 25transforms the equation for the ensemble average of i i) into the formNow, according to the definition of the Laplace transform,-

V(8) =Lm run Hence,

f (T) -(n-Zr) rd VG-27) Combining Equation 26 and Equation 27 results inthe following form of the equation Substituting a=s2r for theintegration variable, theequation takes the form Changing the name ofthe integration variable into .9, transforms the equation into Equation30 is the key formula on which many practical devices may be based. Thisequation states that the ensemble average of the current in lead B ofthe deviceshown in. Fig. 3a is the same as the output current for anunswitched network excited by the same voltage, re duced by the factorA, if the unswitched network is chosen in such a manner that itstransfer admittance is related to the transfer admittance Y(s) of theswitched network by Y+ (s)=Y(s -|-2r).

In particular, when the voltage generators furnish di rect currentvoltages +V and -V, then Equation 30 then assumes the form For largevalues of the time t, it can be shown that lim L Y s+2T ]=1im[8'1-Y(8+2T)]=Y(2T) H 8 8%0 s so that 1-H 2 The meaning of the termlarge values of the time t is that transients that are caused by theswitching on of the generators (not transients that are caused by theswitching, since these are essentially to the operation) have had timeto die out. Consequently, from now on the assumption is made that thosetransients have died out, so that Equation 33 may be rewritten as i(t)=i Combining this with Equation 34, it can be seen that mgr 21 as) FromEquation 35, it can be seen that a measuring apparatus can beconstructed, the output current of which depends on the rate ofoccurrence of input pulse to the network. A condition for constructing ameasuring apparatus having these characteristics is that the inputpulses to the network are alternately positive and negative. Thus, byconnecting an electronic switch which is sequentially switched topositive and negative states by successive pulses representing a timeseries the above requirement is satisfied. It is also apparent fromEquation 35 that the transfer admittance of the network is a function ofa complex frequency which, in the instant case, represents a pulse rate.Consequently, if the network is constructed to have a transferadmittance which is a desired and specific function of the complexfrequency, it will produce an output current which in a similar fashionis functionally related to the input pulse rate.

It has also been shown that in the generalized apparatus of Fig. 3 thatthe output current is positive when the last throw of the right handswitch 44 was positive, and is negative when the last throw of theswitch 44 was to the minus position. Consequently, the right hand switch44 of Fig. 3 may be replaced by a pair of oppositely poled diodes suchas are shown in the specific embodiment of Fig. 2. Similarly, the lefthand switch of Fig. 3 may be replaced by an electronic switch, such as abi-stable multivibrator, which is actuated by pulses coincident with theevents in the time series and function to switch the equilibriumcondition alternately to a positive and negative state. Thus, it can beseen that the generalized apparatus of Fig. 3 may be replaced by simpleelectronic equipment to produce a counting rate apparatus, such as shownin Fig. 2, embodying the instant invention.

As was pointed outwith reference to Figs. 1 and 2, it is desirable formany purposes, such as nuclear instrumentation, to provide a countingrate measuring apparatus which has a logarithmic relationsip between anoutput electrical signal, such as a current, of the apparatus and theinput pulse rate. That is, it is desired that the output depend on therate r according to the equation =%1 1+m (so where R and T are designedparameters. This may be achieved by constructing a two-terminal pairnetwork of the type shown in Fig. 1, characterized by a transferadmittance such that the current flowing in its output is ap'proximately that defined by Equation 36. Thus, according to Equation 35a network must be synthesized which has a transfer admittance Y(s) as afunction of the complex frequency s as given by the equation Y(s log(1+%') 37 This equation is of the form Y(s) g(z) with Z=8T (38) In theinstant case this can be rewritten as z (1+5) (so It will now be shownhow it is possible to synthesize very simple networks for a wide classof functions g(z). This class of functions has the property that g(z)may be expressed in the form where sc 0 and. h(:c) 0 for 2:22; (40)Obviously g(z) must fulfill certain conditions so as to be expressed inthe form disclosed in Equation 40. However, it is not necessary to spellout these conditions in advance. They become clear as one attempts tosolve for the function h(x). After the function h(x) has been found, itis possible to select the proper circuit elements, resistors, capacitorsand inductors, in order to achieve the desired result.

The necessary first step is to find the function h(x). This may be doneby means of well-known procedures utilized in the theory of analyticalfunctions (the transfer admittances are analytical functions of thecomplex frequency). Starting with the Cauchy integral form where theintegration contour enclosures, in a counterclockwise direction, thepoint z but no singularity of an I If the functions g(z) are restrictedto a type whose only singularity lies on the negative real axis to theleft of some point Z 0, the following integration contour may be usedadvantageously.

13 contour'consists. ofa circular portion and a hairpin portion. If;furthermore, g('z) fulfills the condition then, as the radius of thecircular portion is made larger and larger, the contribution to theintegral of Equation 41 from thecircular. portionoithe contour tends tobe 0. Thus, Equation 41 assumes the form 1 z 1 3? lbelow Where()-]above:

is the value of the multivalued function g() on the upper leg of theshrunken hairpin and Where [8) ]below is the value of the multivaluedfunction g() on the lower leg of the shrunken hairpin.

Making: the substitution. =x for the integration variable, and denoting-2 by x ('x 0),vthe;equation takes the: following form In the particularexample shownin Equation 39,.itcan beseen that-'z =2, hence 76 :2, and

consequently h(x)'=% V 1-4 and' Thus; Mac); has turnedoutto. be positivefor'x x as-is clearly shown by Equation 47. At this point itis-neces.sary. to check Equation 48.. It can be seen that; Equation 48, uponsubstitution of the proper functions therein, takes the form A (2+ 2) zi l? 1+2 It" is obvious, once the. substitution is made, that Equation48 devolves into the desirable form exhibiting logarithmiccharacteristics.

In order to facilitate the'synthesis of the network, it is desirable tomake the following substitution x=2+2a dx=2(log a)a dn in the integralof Equation 49. From this substitution Equation 49 takes the form Theintegral of' Equation 50 maybe approximated" by a sum, which thuspermits the obtaining of an approximate value Y s) for the transferadmittance. Thus the equation. for; the. approximate: value Y fi) is 2log a 2 8 T 2a R s T+2+2a 2+2a or, rewriting the above equation, ittakes the form This equation. may then be rewritten. in. the following.form then with the terms R andC being defined by the following equationsEquation 53 shows that the transfer admittance is the admittance of acircuit with an infinite number of parallel branches, n being the ordernumber of the branches,

with. branch n consisting of a resistor with resistance R,

and a capacitor with capacitance. C connectedin, series.

In the limit a- 1, the approximation for the transfer admittance becomesexact, as is shown by the following equation There arepracticaladvantages, relatingto the. number of. parallel branchesrequired, inchoosing. the number, a;

which is a design parameter, to be quite a bit larger than '15 unity.The advantage to be gained is that the larger a the fewer branches arenecessary if a specified range of rates is to be covered. Consequently,it is necessary to determine how well Y (s) approximates Y(s).

For the purpose of a counting rate meter it is necessary to examine Y,(s) and Y(s) for s=2r, where r is the rate of events. Thus, comparingthe transfer admittance Y(s) and the approximation thereof, thefollowing relationship may be seen r T a It is now necessary to compareY -(2r) for two rates r and r related by l+r T=a(l-{rT) The equation forthe approximate value of the transfer admittance is defined as follows:

Y (w,. s a 1+rT 1 1 H R 5,, a(l+rT)+a= 1+a 2loga g l-l-rT 1 :l

R 1+1'T+a 1+0.

or, on changing the summation index from n to n+1 2 log a 1+T 1 rmnr('1) R gm 1+ T+ n 1+an+1] Combining Equations 57 and 56 provides thefollowing transformation In the last sum all terms cancel except theterms However, since a l, the following relationship between thetransfer admittances for the two rates may be established uma) pnr( R ga Furthermore, it can be seen from Equation 55 that the followingrelationship also exists Combining Equation 5 8 and 59 provides thefollowing equation appr( 1)' 1)= appr( This equation shows, that if Yflr) tracks Y(2r) rather 16 well over the range 1+rT=l to 1+rT==a, itwill track also over the range 1+rT=a to 1+rT=a and over the range1+rT=a to 1+rT=a etc. Hence, the approximation need only be examinedover the range 1+rT=l to l+rT=a. For the choice 11:10 the tabulationexhibits the following characteristics:

14mm -Y(2r) It can be seen that, even with a value of a as large as 10the tracking is excellent. The largest deviation between is only .002.Of course, when the design parameter a is chosen closer to unity, thetracking will be even better. However, as was pointed out previously, inthat case larger numbers of the parallel branches will have to beutilized.

In order to illustrate a practical example of such a logarithmiccounting rate apparatus, it may be assumed, for the purpose ofillustration, that it is desired to measure an rT which covers the rangefrom 0 to 10 Ideally, the output current would be defined by theequation V l ldenll 2 g However, let it be assumed that an accuracydefined by the formula is satisfactory, Consequently, it will suffice tolet n, the number of parallel branches in the network, run from X -2 to+6. Thus, nine resistance capacitance combinations will be required forthis desired accuracy. Furthermore, assuming that the following choicesfor the other parameters are satisfactory:

V=23.0 volts T=1 sec. R=1.152 meg.

the output current is approximately i=20 microamps log(1+rT) For therange rT=0 to 10 the counting rate covers the range r=0 to 10 sec- Thecurrent for r=l0* see is then i=20 microamps log (10,001)=184 microamps.The values of the resistances and capacitances for our choices of thedesign parameters R, T, and :1 become i=20 microamps log(1+r sec) towithin 0.2 microamps.

It is obvious, of course, that there is a wide range of choice in thethree design parameters R, T, and a, so

that counting rate meters with logarithmic outputs may be designedaccording to a wide range of specification.

In the mathematical analysis made above, it has been shown by Equation35 that counting rate devices may be designed having output currentswhich are a specified function of an input pulse rate. Furthermore, anetwork was synthesized for use in such a device which provides anoutput current which is a logarithmic function of the input pulse rate.It must be realized, however, that many other networks having functionalrelationships other than logarithmic may be synthesized for use indevices of this type.

For example, it is possible to synthesize a network, and a counting ratemeter, which is characterized by the fact that the transfer admittanceand the output current is related to the rate r by the equation t- (2rT)(61) where R, T, a are design parameters, and a is restricted to thevalues O zx 1. In this instance the current is proportional to afractional power of the rate r. According to Equation 35 there must besynthesized a network having a transfer admittance defined by theequation This network will also be constituted of a number of parallelbranches, each of which comprises series connectedresistance-capacitance combination. In this particular case the functiong(z), as defined in relation to Equation 38 takes the form The branchpoint 2 is now Equation 46 may now be utilized to find the functions gabove and below, thus Thus, utilizing Equation 40 the term Zu may be:determined,

It is again necessary to check the above equation in order to determinewhether it takes "the proper form. It can be seen, on substituting theproper functions therein that Making the following substitution in theabove equation,

L Y d dY z 1Y z (1-Y) it is transformed into the following form Thelatter integral is well-known, and there may be substituted thereforethe following equation 1 f Y (1-- Y) dY=1(a)I(1-a) Y=0 where 1 denotesthe gamma function. Furthermore, it is well-known that On making thefollowing substitution,

x=a dx= (loga) a dn Equation 67 takes the form sT sT+a Y(s) R T a dn(68) Again it is possible to approximate the above integral by thefollowing sum Equation 70 shows that the transfer admittance is theadmittance of a circuit with an infinite number ofparallel branches, nbeing the order number of the branches, with branch n consisting of aresistor with ,a resistance R and a capacitor with a capacitance qconnected ,in

19 series. Equation 71, which defined the magnitude of the resistanceand capacitance components is utilized in designing the variouscomponents of the network. The parameter a therein is a dimensionlessnumber which is greater than and less than 1, while R is a designparameter.

In a manner similar to the two examples disclosed here, many other typesof networks may be synthesized in order to produce a counting ratedevice which produces an output signal which is a desired function ofthe input pulse repetition rate. The mathematical analysis principle ofthis invention has been based on a repetition rate of the input pulsethat is random. However, it can be shown mathematically that theapparatus designed by operation with random series may be utilized aswell for other time series, such as the periodic. It can be shown thatthe current produced by the application of a periodic series will exceedthat produced by the application of a random series. However, for highcounting rates, the excess becomes constant and may, therefore, becornpensated for by means of calibration techniques. Consequently, forhigher counting rates, the apparatus here disclosed may be utilized withperiodic time series, as well as with stationary random series.

It will be appreciated from the foregoing discussion that it is possiblewith the teaching of this invention to construct counting rate measuringdevices having many different types of response, where the term responserefers to the functional relationship between the output current and theinput pulse rate. However, it should also be understood that it ispossible with the teaching of this invention that any counting rateapparatus may be converted into an anti-counting rate apparatus. Thatis, a device that converts a current into a pulse rate having a desiredfunctional relationship to the current. Broadly speaking, such ananti-counting rate device may be constructed by combining a source ofvariable pulses and a counting rate device.

Referring now to Fig. 5, there is shown an anti-counting rate apparatusembodying the principle of the instant invention. There is provided apulse source 53 having a variable pulse rate. The pulse source 53 may beany one of the many well known types which have a variable pulserepetition rate. One type of pulse source which may be advantageouslyutilized in this circuit is the random pulse generator of the radiationdetector type. That is, a radiation detector of the proportional counteror scintillation type is positioned in a constant magnitude radiationfield. The output of the detector is coupled to a discriminating tube,which functions to control the number of pulses per unit time passed toits output circuit.

The discriminating circuit includes a grid biased triode, with themagnitude of the grid bias determining the number of pulses per unittime which are passed. By controlling the magnitude of this grid bias inresponse to a control signal, it is possible, therefore, to vary thenumber of pulses per unit time produced by this pulse producing circuit.Reference is made to Patent No. 2,662,188, issued to K. C. Crumrine etal. for a typical showing of a circuit of this type.

The output from the pulse source 53 is connected to an output terminal52 and to the input of a pulse rate measuring apparatus 54 of the typeillustrated in Fig. 2. The counting rate apparatus 54 includes a network60 having a transfer admittance which is a desired function of a complexfrequency, where this complex frequency represents the input pulse rate.Coupled to the input of the network 60 is an electronic switch means 57,of the bi-stable multi-vibrator type, which provides alternatelypositive and negative pulses in response to the input pulses from thepulse source 53. Coupled to the output of the network 60 are a pair ofoppositely poled diodes 63 and 64.

, The bi-stable multivibrator 57 consists of two alternately conductingspace discharge devices 58 and 59 of the vacuum triode type, which havetheir anodes and control grids cross-coupled by means of the parallelresistance, capacitance networks 61 and 62. Circuits of this type arecharacaterized by the fact that they possess two condi tions of stableequilibrium. The multivibrator remains in one of the other of these twoconditions until some action occurs which causes the non-conductingtubes to conduct. The tubes then reverse their function and remain inthe new condition until another action occurs.

In order to change the equilibrium state of the multivibrator 57, thepulses from the pulse source 53 are injected into the circuit by meansof the double diode 56. The pulses are applied through the diode 56 tothe control grid of the discharge devices 58 and 59. The negative pulseswill, as was explained with reference to Fig. 2, cause the conductingtube to be cut off while the nonconducting tube becomes conducting. Thiscondition then continues until the next negative pulse, at which timethe equilibrium condition will again be reversed. Thus, the electrondischarge devices 58 and 59 are successively and alternatively actuatedto conducting and non-conducting states by successive input pulses. As aresult, the anode voltage of the electron discharge divice 59 issuccessively at a maximum level when the discharge device isnon-conducting and at a minimum level when it is conducting, thuseffectively applying alternately positive and negative voltages of thetype shown in Fig. 46, to the network 60.

Coupled to the output of the counting rate measuring apparatus 54 is acomparison means, comprising a capacitance C wherein the output currentfrom the measuring apparatus 54 is compared with the input currentsupplied through terminal 51 to provide a control voltage on capacitanceC for varying the repetition rate of the pulse source 53 until the twocurrents are equal. If the output current from the counting rateapparatus 54 does not equal the input current applied to the terminal51, there will be produced a voltage on capacitor C which is applied bymeans of leads 65 to the input of a direct current amplifier 55.Although in the embodiment of Fig. 5, a capacitance C has been shown forthe sake of simplicity, it will be obvious that many other suitablecomparison means may be utilized. For example, a differential amplifiersystem of the type disclosed in Waveforms, Chance, Hughes, MacNichol,Sayre, and Williams, McGraw-Hill Book Co., New York, 1949, vol. 19,Radiation Laboratory Series, page 642, Fig. 18.13, may be utilized inplace of the capacitance C.

The output of the direct current amplifier 55 is in turn connected tothe pulse source 53 in order to vary its repetition rate until theoutput current from the counting rate apparatus 54 equals the inputcurrent at the terminal 51. When this equality between the two currentsis achieved, the repetition rate of the pulse source appearing at theoutput terminal 52 is functionally related to the input current in amanner depending on the specific character of the network 60 in thecounting rate apparatus 54. For example, if the network 60 is of thetype which produces an output current which is a logarithm of the inputpulses, the repetition rate of the output pulses at the terminal 52 willbe exponential function of the current applied at the input terminal 51.By examining the equation:

it can be seen that the rate r is an exponential function of the inputcurrent I, that is,

In a similar fashion, by utilizing networks having characteristics otherthan logarithmic, it is possible to produce at the output terminal 52pulses having repetition rates which are different functions of theinput current.

Demonstrating the broad scope of the inventive concept, is the fact thatit is possible by utilizing the various pieces of apparatus disclosedabove, i.e. the anti-.counting rate apparatus, to produce a generalclass of function generators which will convert an input current into anoutput current having a desired functional relationship to the inputcurrent. Since the anti-counting rate device contains a network of thetype illustrated in Fig. 1, while the counting rate apparatus alsocontains a network of this type, and since it is possible to producenetwork having many varied types of functional relationships, it can beseen that many varieties of function generators may be constructed incarrying out the teachings of the instant invention.

Fig. 6 illustrates a function generator embodying the principles of theinstant invention. 'Ihisfunction generator consists, broadly speaking,of an anti-counting rate apparatus 70 having an output terminal 72connected to the input of a counting rate apparatus '86 having an outputterminal 96. The primary components of the function generator areoperationally related in that the anti-counting rate apparatus producesan output pulse rate at its output terminal 72, which is a desiredfunction of a current applied to its input terminal 71. These are ap-.plied to the counting rate apparatus 86 and are converted into an outputcurrent appearing at the output terminal 96 which is a specifiedfunction of the input pulse rate. Consequently, the output current atthe terminal 96 is a function of the input current at 71, the specificfunction depending on the type of network utilized both in theanti-counting rate apparatus and the counting rate apparatus.

The counting rate apparatus 70 includes a pulse source 73 having avariable repetition rate which may be adjusted in response to a controlsignal. The pulse source 73 may be of any well known type and,specifically, may be a scintillation detector or proportional countertype discussed in detail with reference to Fig. 5.

Coupled to the output of the pulse source 73 is a counting rate device74, comprising bi-stable multivibrator 7-7 and network 100, which, forthe sake of clarity in distinguishing it from the counting rateapparatus 86, will be referred to as an internal counting rateapparatus.

The internal counting rate apparatus 74 functions to produce an outputcurrent which is a desired function of the repetition rate of the pulsesfrom the source 73. This output current is applied to a comparis n meanscon-,

sisting, in this instance, of a capacitance C where it is compared witha current applied to the input terminal 71. If the current applied tothe input. terminal 71 and the current from the internal counting rateapparatus 74 are not equal, a control signal is applied by means of thelead 85 to the input of the direct current amplifier 75. The output ofthe amplifier 75 is connected to the pulse source 73 and operates toadjust the repetition rate of the pulse source 73 until an equalitybetween the two currents is achieved. When such equality is achieved,the pulse repetition rate of the pulse source 73 appearing at terminal72, is functionally related to the input current in a manner dependingon the type of network utilized in the internal counting rate meter.That is, iffor example, a network having a logarithmic characteristic isutilized, the repetition rate of the output. pulses, as will .be shownlater by a rigorous mathematical analysis, Will be an exponentialfunction of the input current. i

The internal counting rate meter 74 comprises a bistable multivibrator77 which functions to apply alternately positive and negative pulsevoltages to a network in response to successiveinput pulses from thepulse source 73. The bi-stable multivibrator consists of two spacedischarge devices 78 and 79 of the vacuum triode type having theiranodes and control grids cross coupled by means of the parallelresistancercapacitance network 80 and 81. The input pulses from thesource 73 are applied to the bi-stable multivibrator 77 by means of adouble diode 76 having a common cathode and two anodes that areconnected respectively to anodesof the triodes 78 and 79. Themultivibrator 77 is characterized by two conditions of stableequilibrium. The circuit remains in one or the other until an inputpulse arrives from the pulse source 73. Upon the occurrence of such apulse, the circuit reverses its condition until the occurrence of thenext pulse. As a consequence, the anode of the triode 79 is alternatelyat a maximum level and a minimum 'level depending on whether the tube isnonconducting or conducting.

Connected to the anode of the tn'ode 79 is a network of the typedisclosed in Fig. 1 which consists of a multiplicity of parallelbranches, each of which consists of a series resistance-capacitancecombination. Connected to the output of the network 82 are a ofoppositely poled diodes 83 and 84. The network 82, as has been pointedout previously, may be constructed to have a transfer admittance whichis a desired function of the complex frequency and which, this instance,represents a pulse rate. Thus, there flows in the output of the networka current which is a desired function of the input pulse rate.Consequently, the output terminal 72 provides output pulses which have arepetition frequency which is a specified function of the input currentapplied to the terminal 71 of the counting rate apparatus 70.

Connected to the terminal 72 is a counting rate apparatus 86 which, asexplained previously, functions to transform these pulses into an outputcurrent which is a specified function of the repetition rate of thepulses. The counting rate apparatus 86 similarly includes a bi-stablemultivibrator 88 having coupled to its output a network 87 whosetransfer admittance is a desired function of the input pulse rate. Thebi-stable multivibrator 88 comprises two space discharge devices 89 and90 of the vacuum triode type, having their anodes and control gridscross-coupled by means of the parallel resistance-capacitance networks91 and 92. The input pulses from the terminal 72 are applied to thebi-stable multivibrator 88 by means of a double diode 93 having itsanode members connected respectively to the anodes of the triodes 89 and90. The input pulses function to reverse the equilibrium conditions ofthe circuit upon the occurrence of each input pulse. Consequently, thevacuum triode 89 is successively placed in a conducting andnon-conducting condition by these pulses, and its anode voltage isalternately at maximum and minimum level depending on its conducting ornon-conducting conditions. These effectively alternate positive andnegative pulse voltages are applied to the network 87 and will cause acurrent to flow which is a specified function of the pulse rate.

Coupled t0 the output of the counting rate apparatus 86 is anadjustableconstant current source 97. This constant current sourcefunctions either to add a constant DC. current of a given magnitude orto subtract a constant DC current of a given magnitude. The function ofthis constant current source will be explained in greater detail when amore rigorous mathematical analysis of the operation of the functiongenerator is given. The constant current source may be any of many wellknown types. For example, it could constitute a battery, or a constantcurrent pentode, or even a regulated current source. The specificconstruction of the constant current source is not critical as long asthe magnitude of the current produced thereby is both constant andadjustable.

As has been stated previously, the pulse rate r appearing at the outputterminal 72' is a function of the input current I applied at theterminal 71. The precise function depends on the design of the network82 in the internal counting rate apparatus 74, Forexample, if. a networkis synthesized which provides a logarithmic res sponse, then therelationship between the current and rate of pulses is defined by theequation:

23 Thus, it can be seen that the repetition rate of the output pulses isan exponential function of the input current 1. Suppose that for thecounting rate apparatus 86 a network is utilized that has a linearscale, so that its output current I follows the equation:

where A is a design parameter. Consequently, the output current may bedefined in terms of the input current I by the equation:

A RI li'e p 7 If a constant DC. current of magnitude is added to theoutput current I, then the current appearing at the output terminal 96may be defined by the equation:

Thus, there has been constructed a function generator which produces acurrent K that is an exponential function of an input current I.

In order to demonstrate the extreme flexibility of the instant inventionin constructing function generators having many sorts of functionalrelationships between the input current and the output current, assumethat the network 82 in the internal counting rate apparatus 74 has alinear scale and the network 87 of counting rate apparatus 86, has a-logarithmic scale. Thus, the relationship between the input current Ito the anti-counting rate device 70 and the pulse rate is defined by theequation:

I=Ar or 1-:

Since the output current I may be defined by the equation:

1 J log (1+TT) this equation in its final form states,

Suppose that from some given current H there is subtracted a constantcurrent of a magnitude If the difference of these two currents is thenapplied to the combination of anti-counting rate apparatus and countingrate apparatus, the input current I to the terminal 71 takes the form:

As a consequence, the output current I may now be defined by theequation:

In this manner, it has been possible to construct a func- 24 tiongenerator that produces an output current I which is a logarithmicfunction of an input current.

From these examples, it should be obvious that a very large class offunction generators may be constructed embodying the principles of theinstant invention. The precise functional relationship of the input tothe output current which may be created will depend on the character ofthe networks 82 and 87 which are utilized. Although the networks whichhave so far been illustrated and discussed, have contained resistanceand capacitance elements, it is obvious, of course, that reactiveelements other than capacitances may be utilized. That is, thenetworkmay be constructed of resistive and inductive elements, for example, andstill fall within the scope of this invention. It is also possible touse series inductive-capacitive combinations as along as the circuitsare such that the current pulse illustrated in Fig. 4b eventuallyreaches a steady state condition.

Furthermore, although the components of these networks have been shownas passive elements, so that the network as a whole is passive, theinventive concept is not limited to passive networks but may incorporatetherein active elements such as amplifiers. It should also be pointedout that although the previous discussion has been carried out in termsof transfer admittances which describe relationships between inputvoltages and output currents, it is quite obvious that the instantinvention is not limited thereto. That is, the voltage sources v(t) and-v(t), illustrated in Fig. 3a, may be replaced by constant currentsources i(t) and i(t) and a network synthesized having the desiredtransfer impedance.

While I have shown a particular embodiment of this invention it will, ofcourse, be understood that the invention is not limited thereto sincemany modifications both in the circuit arrangement and in theinstrumentalities employed may be made. It is contemplated by theappended claims to cover any such modifications as fall within the truespirit and scope of this invention.

What I claim as new and desire to secure by Letters Patent of the UnitedStates is:

l. A pulse rate measuring apparatus, comprising means actuated inresponse to input pulses having a variable rate of occurrence forfurnishing successive positive and negative pulses, network meanscoupled to said last named means, said network means characterized by anadmittance which is a function of a complex frequency representing apulse rate so that an output signal is produced whichis dependent on therate of occurrence of said pulses, and unidirectional conducting meanscoupled to the output of said network.

2. A pulse rate measuring apparatus, comprising switch means actuated inresponse to pulses having a variable rate of occurrence, said switchmeans adapted to furnish successive positive and negative pulses inresponse to successive input pulses, network means coupled to saidswitch means characterized by an admittance which is a function of acomplex frequency representing a pulse rate so that an output signal isproduced which is dependent on the rate of occurrence of said pulses,and unidirectional conducting means coupled to the output of saidnetwork.

3. A pulse rate measuring apparatus, comprising switch means actuated inresponse to random pulses having a variable rate of occurrence, saidswitch means adapted to furnish successive positive and negative pulsesin response to successive input pulses, network means coupled to saidswitch means for receiving said successive positive and negative pulses,said network being characterized by a transfer admittance which is alogarithmic function of a complex frequency representing a pulse rate sothat a current flows which is a logarithmic function of the rate ofoccurrence of said input pulses, and unidirectional conducting meanscoupled to the output of said network.

4. A pulse rate measuring apparatus, comprising switch means actuated inresponse to random pulses having a variable rate of occurrence, saidswitch means adapted to, furnish successive positi e; and negativepulses in, re; sponse to successive input pulses, network meanscoupled,tosaid switch means for receiving said successive positive and; negativepulses, said network being characterized by a transfer-- admittancewhich is. a fractional power {1111C} tionof a; complex frequencyrepresenting a pulse rate sothata; current flows which is a fractionalpower function, of the rate of-occurrence of said input pulses, and unidirectional conducting means coupled to the output of; said; network. 7

5,. A pulse ratemeasuring-apparatus comprising switch means actuated inresponse, to pulses having a; variable rate of occurrence, said switchmeans adapted to furnish successive positive, and negative pulses inresponse to successive input pulses, network means, coupled to, saidswitch means for receiving said successive positive and negative.pulses, said network being characterized by a transfer admittance whichis a logarithmic function of a complex frequency representing a pulserate whereby a current flows which is a logarithmic function of the.rate of occurrence of said pulses, said network comprising amultiplicity ofparallel connected series resistance capacitancebranches, and undirectional conducting means coupled to the output ofsaid network.

6. A pulse rate measuring apparatus, comprising switch means actuated inresponse to random pulses having a variable rate of occurrence, saidswitch means adapted tofurnish successive positive and negative pulsesin: response to successive input pulses, network means. coupled to saidswitch means for receiving said successive positive and negative pulses,said network being characterized by a transfer admittance which isalogarithmic function of a complex frequency representing a pulse rateso that a current flows which is a logarithmicfunction of the rate ofoccurrence of said pulses, said network comprising n parallel connectedseries; resistance-capacitance branches, the magnitudes of; theresistive and capacitive elements-being defined by where r is the ratevof occurrence of the input pulses, T is time, and a and R are designedparameters, and unidi rectional conducting means coupled to the outputof said network;

7. A pulse. rate measuring apparatus, comprising electronic switch meansactuated in response to random pulses, having a variable rate ofoccurrence, said switch meansadapted to furnish successive positive andnega-v tive pulses in response to successive input pulses, network meanscoupled to said switch means for receiving said positive and negativepulses, said network beingcharacterized by a transfer admittance whichis a function of a complex frequency repr enting a pulse rate so that acurrent flows; which is dependent upon the rate of occurrence of saidinput pulses, aunidirectional conducting means coupled to the output ofsaid network.

8. A pulse rate measuring apparatus, comprising electronic switch meansactuated in response to random pulses having a variable rate ofoccurrence, said switch means adapted to furnish successive positive andnegative pulses in response to successive input pulses, network meanscoupled to said switch means for receiving said PQSitive and negativepulses, said network being characterized by a transfer admittance whichis a loga rithmic function of a complex frequency representing a pulserate. so that a current flows which is a logarithmic 26 nd paral e a cone ed un di a cond c n coupled to the output of said network.

9; A pulse rate measuring apparatus, comprising a bias-tablemultivibrator actuated in response to -succes-; sive input pulses totransfer its conductive states and furnish successive; positive andnegative pulses, saidine put pulses having a variable and randomrate-ofioccur rence, network means coupled to said multivibrator-forreceiving said positive and negative pulses, said network; beingcharacterized by a transfer of admittance which is a logarithmicfunction of a complex frequency rep-.- resenting a pulse rate so that acurrent flows which is a logarithmic t'unction of the rate of occurrenceof said input pulses, said network comprising a multiplicity of seriesresistance-capacitance combinations connected in parallel, oppositelypoled rectifying means connected to said network.

10. A pulse rate measuring apparatus, comprising; a bi-stablemultivibratoractuated successively to transfer its conductive states tofurnish successive positive and negative pulses in response tosuccessive input pulses, said input pulses having a random and variablerate of occurrence, rectifying means connected to the input of saidmultiyibrator to'apply said pulses thereto, network means coupled to theoutput of said multivibrator to receive said positive and negativepulses, said network being characterized by a transfer admittancewhichis a logarithmic function of a complex frequency representing apulse rate so that a current flows which is a logarithmic function of;the rate of occurrence of said input pulses, said network comprising amultiplicity of series resist} antic-capacitancecombinations connectedin parallel, and oppositely poled, diodes connected to said network.

' 11. Inan apparatus for generating pulses whose repetition rate is afunction of an input current, the combina tion comprisingpulsegenerating means having a variable repetition rate, counting rate meanscoupled to said pulse generating means for producing a current which a fc o i h r pe i io r t of a Pu in l ins a swi ch ns a t n po e to pu se rm said generating means to furnish successive positive, and negative.pulses, a network coupled to said switch, means for; receiving saidpositive and negative pulses, said net: work having an admittance thatis, dependent on the pulse repetition rate for producing an outputcurrent which is a functionv 0f the. pulse repetition rate, comparisonmeans having applied thereto current from saidcountng ate. means, andsaid input current, and means to vary the repetition rate of said pulsegenerating means until said currents. are equal whereby the repetitionrate, of saidpulse generating means is a function said input current.

'12,. In an apparatus for generating pulses Whose rcpefi; tion rate is afunction of an input current, the combina: tion comprising pulsegenerating means having a variable repetition rate, counting rate meanscoupled to said pulse generating means for producing a current which isa function of the repetition rate of said pulses including a switchmeans, actuated in response. to pulses from, said generating means tofurnish successive positive and negative: pulses, a network coupled tosaid switch means for receiving said positive and negative pulses,.saidnetwork being characterized .by a transfer admittance that is dependenton the pulse repetition rate for producing an output current which is afunction of the pulse repetition rate, comparison means including astorage means having applied thereto the current from said counting ratemeans and the input current, means to vary the repetition rate of saidpulse generating means until said cur-rents are equal whereby therepetition rate of said pulse generating means is a function of saidinput current.

13. In an apparatus for generating pulses whoserepcti; tion rate is afunction of an input current, the combination,

function of the rate of. occurrence of said inpu rulsea 76 comp is ngPul e nera g me n having a afiablerqpg;

27 tition rate, counting rate means coupled to said pulse generatingmeans for producing a current which is a logarithmic function of therepetition rate of said pulses including a switch means actuated inresponse to pulses from said generating means to furnish successivepositive and negative pulses, a network coupled to said switch means forreceiving said positive and negative pulses, said network beingcharacterized by a transfer admittance which is a logarithmic functionof a complex frequency representing said pulse rate for producing anoutput current which is a logarithmic function of the pulse repetitionrate, comparison means including a capacitance having applied theretothe current from said counting rate means and said input current, meansresponsive to the output from said comparison means to vary therepetition rate of said pulse generating means until said currents areequal whereby the repetition rate of said pulse generating means is anexponential function of said input current.

14. In an apparatus for generating pulses whose repetition rate is afunction of an input current, the combination comprising pulsegenerating means having a variable repetition rate, counting rate meanscoupled to said pulse generating means for producing a current which isa logarithmic function of the repetition rate of said pulses, saidcounting rate means including electronic switch means adapted to furnishsuccessive positive and negative pulses in response to successive inputpulses from said generating means, and a network coupled to said switchmeans having transfer admittance which is a logarithmic function of acomplex frequency representing said pulse rate for producing an outputcurrent which is a logarithmic function of the pulse repetition rate,comparison means having applied thereto the current from said countingrate means responsive to the output from said comparison means and theinput current, means to vary the repetition rate of said pulsegenerating means until said currents are equal whereby the repetitionrate of said pulse generating means is an exponential function of saidinput current.

15. In an apparatus for generating pulses whose repetition rate is afunction of an input current, the cambination comprising pulsegenerating means having a variable repetition rate, counting rate meanscoupled to said pulse generating means for producing a current which isa logarithmic function of the repetition rate of said pulses, saidcounting rate means including electronic switch means adapted to furnishsuccessive positive and negative pulses in response to successive inputpulses from said generating means, and a network coupled to said switchmeans for receiving said positive and negative pulses, said networkcomprising a multiplicity of parallel connected seriesresistance-capacitance combinations having a transfer admittance whichis a logarithmic function of a complex frequency representing said pulserate for producing an output current which is a logarithmic function ofthe pulse repetition rate, comparison means responsive to'the outputfrom said comparison means having applied thereto the current from saidcounting rate means and said input current, means to vary the repetitionrate of said pulse generating means until said currents are equalwhereby the repetition rate of said pulse generating means is anexponential function of said input current.

16. The apparatus of claim wherein said comparison means includes acapacitance. j

17. In a function generator, the combination comprising means togenerate pulses havinga repetition rate which is a given function of aninput current, means coupled to said pulse generating means to producean output current which is a function of the repetition rate of saidpulses including a switch means actuated in response to pulses from saidgenerating means to furnish successive positive and negative pulses, anetwork coupled to said switch means for receiving said positive andnegative pulses, said network being characterized by a transferadmittance which is a function of a complex frequency representing saidpulse'rate so that a current flows which is dependent 28 on therepetition rate of said pulses whereby the output current is a desiredfunction of the input current.

18. In a function generator, the combination comprising means togenerate pulses having a repetition rate which is a given function of aninput current including a variable rate pulse source, a counting ratemeans to produce a current which is a function of the pulse rate, meansto vary the repetition rate of said pulses until the current from saidcounting rate means equals said input current whereby the repetitionrate is a function of the input current, means coupled to said pulsegenerating means to produce an output current which is a function of therepetition rate of said pulses including a switch means actuated inresponse to pulses from said generating means to furnish successivepositive and negative pulses, a network coupled to said switch means forreceiving said postive and negative pulses, said network beingcharacterized by a transfer admittance which is a function of a complexfrequency representing said pulse rate so that a current flows which isdependent on the rate of occurrence of said pulses whereby the outputcurrent is a desired function of the input current.

19. In a frmction generator, the combination comprising means togenerate pulses having a repetition rate which is an exponentialfunction of an input current, including a pulse source having a variablerepetition rate, a switch means actuated in response to said generatingmeans to furnish successive positive and negative pulses, a networkcoupled to said switch means for receiving said positive and negativepulses, said network being characterized by a transfer admittance whichis a logarithmic function of a complex frequency representing the pulserate and causes a current flow which is a logarithmic function of thepulse repetition rate, means coupled to said pulse generating means toproduce an output current which is a linear function of the repetitionrate of said pulses whereby the output current is an exponentialfunction of the input current.

20. In a function generator, the combination comprising means togenerate pulses having a repetition rate which is a linear function ofan input current including a pulse source having a variable repetitionrate, a switch means actuated in response to pulses from said pulsesource to furnish successive positive and negative pulses, a networkcoupled to said switch means for receiving said positive and negativepulses, said network being characterized by a transfer admittance whichis a linear function of a complex frequency representing said pulse ratewhereby a current flows which is a linear function of the pulserepetition rate, means coupled to said pulse generating means to producean output current which is a logarithmic function of the repetition rateof said pulses whereby the output current is a logarithmic function ofthe input current. v

21. In a function generator, the combination comprising means togenerate pulses having a repetition rate which is the given function ofan input current including a pulse source having a variable repetitionrate, and counting rate means to produce a current which is a functionof the repetition rate including a switch means actuated in response topulses from said generating means to furnish successive positive andnegative pulses, a network coupled to said switch means for receivingsaid positive and negative pulses, said network being characterized by atransfer admittance which is a predetermined function of a complexfrequency representing the pulse rate whereby a current flows which is apredetermined function of the input pulse repetition rate, means tocompare the current from said counting rate means and the input current,means responsive to the output from said comparison means to vary therepetition rate of said pulses until the current from said counting ratemeans equals the input current whereby the repetition rate is a functionof the input current, means coupled to said pulse generatingmeans toproduce an output current which is 29 a function of the repetition rateof said pulses including a network characterized by :a transferadmittance which is a function of a complex frequency representing saidpulse rate so that an output current dependent on the repetition rate ofsaid pulses flows whereby the output current is a desired function ofthe input current.

22. The apparatus of claim 21 wherein said counting rate means includesa network characterized by a transfer admittance which is a function ofthe repetition rate of the pulses produced by said pulse source.

References Cited in the file of this patent UNITED STATES PATENTSDietzold Apr. 17, 1951 Leste June 12, 1951 Lovell Nov. 3, 1953Philbric'k Jan. 10, 1956 Lilienstein July 8, 1958

